The concept of a moduli space goes back to Riemann's count of $3g-3$ (or $3p-3$, in older notation) coordinates to describe Riemann surfaces of genus $g$ when $g > 1$. See the bottom of p. 33 here, where Riemann introduces the term Moduln and writes multiple times on later pages about Anzahl der Moduln (number of moduli/parameters/coordinates). This was in 1857.

The 19th century gave rise to other similar-sounding terms: modular functions, modular forms, and the modular group $\text{SL}_2(\mathbf Z)$. As far as I am aware, these terms were introduced in work of people like Dedekind, Fricke, and Klein in the 1870s-1890s, and is related to Dedekind's creation of the term Modul for a lattice (in Euclidean space). Lattices in $\mathbf C$, after suitable scaling normalization to the form $\mathbf Z + \mathbf Z\tau$, are closely related to $\text{SL}_2(\mathbf Z)$, and modular functions and modular forms are related to the action of this group on the upper half-plane (and modular forms can be thought of as certain homogeneous functions on the space of lattices).

My question: is there any reason (through inspiration perhaps, if not for precise technical reasons) to expect Dedekind's term was inspired specifically by Riemann's term?

This question is an attempt to get some clarification on my answer to the MathOverflow question here.


1 Answer 1


Dedekind explained his choice of "modul" in a footnote with reference to Gauss's congruences, not to Riemann's work, see Episodes in the History of Modern Algebra (1800-1950), p. 79. In 1863 Dedekind published posthumously Dirichlet's Lectures on Number Theory, and Dirichlet defined for $a=sk+r$, , see Siebeneicher's Residues: The gateway to higher arithmetic:

"In what follows we shall say that $r$ is the remainder of the number $a$ relative to the modulus $k$".

Dedekind's point apparently was that the congruence relation is determined by a set, a subset of $\mathbb{Z}$, just as his lattice moduls are subsets (additive subgroups, in modern terms) of $\mathbb{C}$.

Riemann's work on complex surfaces was clearly an inspiration for many algebraists including Kronecker and Dedekind-Weber. Dieudonné in Historical Development of Algebraic Geometry (American Mathematical Monthly, 79 (1972) no. 8, 827-866) writes, for example:

"The goal of Dedekind and Weber in their fundamental paper was quite different and much more limited; namely, they gave purely algebraic proofs for all the algebraic results of Riemann."

The "fundamental paper" is Theory of Algebraic Functions of One Variable, and 1882 is after Dedekind already introduced his modul and published Dirichlet's lectures with an independent use of "modulus". If anything, Riemann's use of "modulus" might have been "inspired" by Dirichlet's, but in a purely linguistic sense, the word was around and its Latin root fit.


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